Generalized Beatty sequences

Abstract

A well-known result due toS. Beatty is that ifαandβare positive irrational numbers satisfyingα−1+β−1=1then each positive integer is to be found in precisely one of the sequences{[kα]},{[kβ]}(k=1,2,3,…)where[x]denotes the integral part ofx. The present note generalizes this result to the case of the pair of sequences{[f(k)]},{[g(k)]}with suitable hypotheses on the functionsfandg. The special casef(x)=αx,g(x)=βxis the result due to Beatty.